Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions

The Wagner [1] and Smith [2], [3] classical monostatic one-dimensional (1-D ) shadowing functions assume that the joint probability density of heights and slopes is uncorrelated, thus inducing an overestimation of the shadowin g function. The goal of this article is to quantify this assumption. More r ecently, Ricciardi and Sate [4], [5] proved that the shadowing function is given rigorously by Rice's infinite series of integrals, We observe that th e approach proposed by Wagner retains only the first term of this series, w hereas the Smith formulation uses the Wagner model by introducing a normali zation function. In this article, we first calculate the shadowing function based on the Ricciardi and Sate work for an uncorrelated process. We will sec that the uncorrelated results do not have any physical sense. Next, the Wagner and Smith formulations will be modified in order to introduce the c orrelation. Correlated and uncorrelated results are compared with the refer ence solution, which is determined by generating a surface [8] for a Gaussi an autocorrelation function, So, we will show that the correlation improves the results for values mu less than or equal to 2 sigma, where mu represen ts the slope of incident ray and sigma the slopes variance of the surface. Finally, our results will be compared to those given in [9], determined fro m the first three terms of Rice's series, but the shadowing function used i s not averaged over the slopes.